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Register a new userThe spectrum of perturbations inside the Schwarzschild black hole
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Authors
Hassan Firouzjahi
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We study the spectrum of the bound state perturbations in the interior of the Schwarzschild black hole for the scalar, electromagnetic and gravitational perturbations. Demanding that the perturbations to be regular at the center of the black hole determines the spectrum of the bound state solutions. We show that our analytic expression for the spectrum is in very good agreement with the imaginary parts of the high overtone quasi normal mode excitations obtained for the exterior region. We also present a simple scheme to calculate the spectrum numerically to good accuracies.
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We study linear gravitational perturbations of Schwarzschild spacetime by solving numerically Regge-Wheeler-Zerilli equations in time domain using hyperboloidal surfaces and a compactifying radial coordinate. We stress the importance of including the asymptotic region in the computational domain in studies of gravitational radiation. The hyperboloidal approach should be helpful in a wide range of applications employing black hole perturbation theory.
We present the first numerical construction of the scalar Schwarzschild Green function in the time-domain, which reveals several universal features of wave propagation in black hole spacetimes. We demonstrate the trapping of energy near the photon sphere and confirm its exponential decay. The trapped wavefront propagates through caustics resulting in echoes that propagate to infinity. The arrival times and the decay rate of these caustic echoes are consistent with propagation along null geodesics and the large l-limit of quasinormal modes. We show that the four-fold singularity structure of the retarded Green function is due to the well-known action of a Hilbert transform on the trapped wavefront at caustics. A two-fold cycle is obtained for degenerate source-observer configurations along the caustic line, where the energy amplification increases with an inverse power of the scale of the source. Finally, we discuss the tail piece of the solution due to propagation within the light cone, up to and including null infinity, and argue that, even with ideal instruments, only a finite number of echoes can be observed. Putting these pieces together, we provide a heuristic expression that approximates the Green function with a few free parameters. Accurate calculations and approximations of the Green function are the most general way of solving for wave propagation in curved spacetimes and should be useful in a variety of studies such as the computation of the self-force on a particle.
We formulate and solve the problem of spherically symmetric, steady state, adiabatic accretion onto a Schwarzschild-like black hole obtained recently. We derive the general analytic expressions for the critical points, the critical velocity, the critical speed of sound, and subsequently the mass accretion rate. The case for polytropic gas is discussed in detail. We find the parameter characterizing the breaking of Lorentz symmetry will slow down the mass accretion rate, while has no effect on the gas compression and the temperature profile below the critical radius and at the event horizon.
We investigate perturbations of a class of spherically symmetric solutions in massive gravity and bi-gravity. The background equations of motion for the particular class of solutions we are interested in reduce to a set of the Einstein equations with a cosmological constant. Thus, the solutions in this class include all the spherically symmetric solutions in general relativity, such as the Friedmann-Lema^{i}tre-Robertson-Walker solution and the Schwarzschild (-de Sitter) solution, though the one-parameter family of two parameters of the theory admits such a class of solutions. We find that the equations of motion for the perturbations of this class of solutions also reduce to the perturbed Einstein equations at first and second order. Therefore, the stability of the solutions coincides with that of the corresponding solutions in general relativity. In particular, these solutions do not suffer from non-linear instabilities which often appear in the other cosmological solutions in massive gravity and bi-gravity.
Linearized perturbations of a Schwarzschild black hole are described, for each angular momentum $ell$, by the well-studied discrete quasinormal modes (QNMs), and in addition a continuum. The latter is characterized by a cut strength $q(gamma>0)$ for frequencies $omega = -igamma$. We show that: (a) $q(gammadownarrow0) propto gamma$, (b) $q(Gamma) = 0$ at $Gamma = (ell+2)!/[6(ell-2)!]$, and (c) $q(gamma)$ oscillates with period $sim 1$ ($2Mequiv1$). For $ell=2$, a pair of QNMs are found beyond the cut on the unphysical sheet very close to $Gamma$, leading to a large dipole in the Greens function_near_ $Gamma$. For a source near the horizon and a distant observer, the continuum contribution relative to that of the QNMs is small.
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